It is well known that minimizers of the Allen-Cahn-type functional \[ J_\epsilon(u):=\int_\Omega\frac{ \epsilon|\nabla u|^2}{ 2}+\frac{ W(u)}{ \epsilon}, \] where $W$ is a double-well potential, resemble minimal surfaces in the sense that their level sets converge to a minimal surface as $\epsilon\rightarrow 0$. In this work, we consider the indicator potential $W(\tau)=\chi_{ (-1,1)}(\tau)$, which leads to the Bernoulli-type free-boundary problem \[ \left\{ \begin{ alignedat}{ 2} \Delta u&=0&\quad&\textrm{ in}\quad\{ |u|<1\}\\ |\nabla u|&=\epsilon^{ -1}&\quad&\textrm{ on}\quad\partial \{ |u|<1\}. \end{ alignedat} \right. \] We provide a short proof that the transition layers are uniformly $C^{ 2,\alpha}$ regular, up to the free boundary. In addition to the uniform $C^{ 2,\alpha}$ estimate, we also obtain improved $C^\alpha$ mean curvature bound that decays in an algebraic rate of $\epsilon$, which confirms the convergence of interfaces to the minimal surface in a very strong sense. We present a simple elliptic equation \[ \Delta\phi=H^2-|\mathbf{ A}|^2 \] where $\phi=\log(1/|\nabla u|)$ is the log-gradient of $u$, $H$ and $\mathbf{ A}$ are the mean curvature and the second fundamental form of level surfaces, respectively. From this, the uniform estimates readily follow. The whole argument is performed in a general Riemannian manifold setting.
Categories: math.AP